Fernique's theorem

Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has exponential tails. The result was proved in 1970 by Xavier Fernique.

Statement
Let (X, || ||) be a separable Banach space. Let &mu; be a centred Gaussian measure on X, i.e. a probability measure defined on the Borel sets of X such that, for every bounded linear functional ℓ : X → R, the push-forward measure ℓ∗&mu; defined on the Borel sets of R by


 * $$( \ell_{\ast} \mu ) (A) = \mu ( \ell^{-1} (A) ), $$

is a Gaussian measure (a normal distribution) with zero mean. Then there exists &alpha; &gt; 0 such that


 * $$\int_{X} \exp ( \alpha \| x \|^{2} ) \, \mathrm{d} \mu (x) < + \infty.$$

A fortiori, &mu; (equivalently, any X-valued random variable G whose law is &mu;) has moments of all orders: for all k ≥ 0,


 * $$\mathbb{E} [ \| G \|^{k} ] = \int_{X} \| x \|^{k} \, \mathrm{d} \mu (x) < + \infty.$$